Inference on Endogenously Censored Regression
Models Using Conditional Moment Inequalities∗
Shakeeb Khana†
a Department
b Department
and Elie Tamerb
of Economics, Duke University , Durham, NC 27708.
of Economics, Northwestern University, 2001 Sheridan Rd, Evanston, IL 60208.
This Version: August 2008
Abstract
Under a quantile restriction, randomly censored regression models can be written in
terms of conditional moment inequalities. We study the identified features of these moment inequalities with respect to the regression parameters. These inequalities restrict
the parameters to a set. We then show regular point identification can be achieved under a set of interpretable sufficient conditions. Our results generalize existing work on
randomly censored models in that we allow for covariate dependent censoring, endogenous censoring and endogenous regressors. We then provide a simple way to convert
conditional moment inequalities into unconditional ones while preserving the informational content. Our method obviates the need for nonparametric estimation, which
would require the selection of smoothing parameters and trimming procedures. Maintaining the point identification conditions, we propose a quantile minimum distance
estimator which converges at the parametric rate to the parameter vector of interest,
and has an asymptotically normal distribution. A small scale simulation study and an
application using drug relapse data demonstrate satisfactory finite sample performance.
JEL Classification: C13; C31
Keywords: Conditional Moment Inequality models, quantile minimum distance, covariate dependent censoring, heteroskedasticity, endogeneity.
∗
We thank K. Hirano, B. Honoré, J. Powell, I. Prucha and A. Rosen as well as seminar participants at
many institutions, the 2005 World Congress meetings of the Econometric Society and the 2006 Econometrics
in Rio conference for many helpful comments. We also thank an editor and the referees for their comments.
Both authors also gratefully acknowledge support from the National Science Foundation. Any errors are our
own.
†
Corresponding Author: Shakeeb Khan, Department of Economics, Duke University, 213 Social Sciences
Building, Durham, NC 27708. Phone: (919) 660-1873.
1
Khan and Tamer 2
1
Introduction
Much of the recent econometrics, statistics, and biostatistics literature has been concerned
with distribution-free estimation of the parameter vector β0 in the linear regression model
yi = x′i β0 + ǫi
(1.1)
where the dependent variable yi is subject to censoring that can potentially be random.
For example, in the duration literature, this model is known as the accelerated failure
time1 (or AFT) model where y, typically the logarithm of survival time, is right censored at
varying censoring points due usually either to data collection limitations or competing risks.
The semiparametric literature which studies variations of this model is quite extensive
and can be classified by the set of assumptions that a given paper imposes on the joint distribution of (xi , ǫi , ci ) where ci is the censoring variable. Under very weak assumptions on this
joint distribution, one strand of the literature gives up on point (or unique) identification
of β0 and provides methods that estimate sets of parameters that are consistent with the
assumptions imposed and the observed data. The bounds approach, explained in Manski
and Tamer (2002), emphasizes robustness of results and clarity of assumptions imposed at
the cost of estimating sets of parameters that might be large2 . The more common approach
to studying the model above starts with a set of assumptions that guarantee point identification and then provides consistent estimators for the parameter vector of interest under these
assumptions. Work in this area includes3 the papers by Buckley and James (1979), Powell
(1984), Koul, Susarla, and Ryzin (1981), Ying, Jung, and Wei (1995), Yang (1999), Buchinsky and Hahn (2001) Honoré, Khan, and Powell (2002) and, more recently Portnoy (2003)
and Cosslett (2004). Some of the assumptions that these papers use are: homoskedastic
errors, censoring variables that are independent of the regressors and /or error terms, strong
support conditions on the censoring variable which rule out fixed censoring, and exogenous
regressors.
We aim to recast the model above in a framework that delivers point identification yet
1
An alternative class of models used in duration analysis is the (Mixed) Proportional Hazards Model.
See Khan and Tamer(2007) and the references therein for recent developments in those models.
2
Recently. Honoré and Lleras-Muney(2006) derive bounds for a competing risk model, which is related
to the randomly censored regression model above.
3
None of these papers impose that ci itself is restricted to behave like a mixed proportional hazards or
accelerated failure time model, as was assumed in point identification results in Heckman and Honoré (1990)
and Abbring and van den Berg(2003). In many settings it is difficult to justify any modelling of the censoring
process, and inconsistent results generally arise when this process is misspecified.
Khan and Tamer 3
weakens many of the assumptions used in the existing literature. In particular, we wish
to construct an estimation procedure that allows for endogenous regressors, the censoring
variable to be endogenous (i.e. correlated with the error terms) and depend on the covariates
in an arbitrary way, and also permits the error terms to be conditionally heteroskedastic.
Our approach to attaining such generalizations will be to write the above censored regression model as a conditional moment inequality model, of the form
E[m(zi ; β0 ) xi ] ≤ 0
(1.2)
where m(.) is a known function (up to β0 ), (zi , xi ) are observed. This class of models has
been studied recently in econometrics. See Manski and Tamer (2002), Chernozhukov, Hong,
and Tamer (2002), Andrews, Berry, and Jia (2003), Pakes, Porter, Ho, and Ishii (2005),
Andrews and Soares (2007), and Rosen (2006). Usually, in conditional moment equality
models, if those moment conditions are satisfied uniquely at a given parameter, then one can
obtain a consistent estimator of this parameter by taking a sample analog of an unconditional
moment condition with an appropriate weight function. The choice of weight functions in
moment equality models is motivated by efficiency gains (Dominguez and Lobato (2004) is
an exception4 ) and not consistency. Other papers that consider moment inequalities are
Andrews and Soares (2007), Canay (2007) and references therein.
With moment inequality models, transforming conditional moment inequalities into unconditional ones is more involved since this might entail a loss of identification. This comes
from the fact that, E[m(zi ; β0 )|xi ] ≤ 0 ∀xi a.e. ⇒ E[w(xi )m(zi ; β0 )] ≤ 0 for an appropriate nonnegative weight function, but generally E[w(xi )m(zi ; β0 )] ≤ 0 6⇒ E[m(zi ; β0 )|xi ] ≤
0 xi a.e. More importantly, in models that are partially identified, the choice of the “instrument function” w(.) affects identification in that different weight function can lead to
different identification regions (as opposed to methods with equality constraints like GMM,
where choice of w generally impacts efficiency, but not point identification). Ideally, one
would want a weight function that leads to the identified (or sharp) set.
One approach to handling moment inequalities while preserving the information content is in Manski and Tamer (2002), who estimated the conditional moment condition nonparametrically in a first step. This is not practically attractive since xi might be multidimensional and nonparametric estimation requires choosing smoothing parameters and trimming
procedures. On the other hand, Pakes, Porter, Ho, and Ishii (2005) use an unconditional
version of the above moment inequality as the basis for inference. Other papers on moment inequality takes a set of unconditional moment conditions as its starting point. In this
4
We thank Adam Rosen for bringing this to our attention.
Khan and Tamer 4
paper, we use insights from Bierens (1990) and Chen and Fan (1999) where a conditional
moment equality is transformed into an unconditional one in testing problems while preserving power. Although their weight functions do not generally apply to models with inequality
restrictions, we will show that a variant of their approach that uses more “localized” weight
functions can be used for conducting inference on the parameters of interest. In particular,
we transform the conditional moment condition to an unconditional one using a class of two
sided indicator functions.
We show under sufficient conditions that the estimator based on this transformation is
consistent and we derive its large sample distribution. While we illustrate this method in
detail in the context of the randomly censored regression model, our approach can be used
for any point identified conditional moment inequality model.
The next section describes the censored model studied in this paper in detail, and establishes the resulting moment inequalities. We then transform the randomly censored model
with conditional inequality restriction into one with unconditional moment inequalities which
will motivate our proposed minimum distance estimation procedure. We then establish the
asymptotic properties for the proposed procedure, specifying sufficient regularity conditions
for root-n consistency and asymptotic normality. Section 5 explains how to modify the proposed procedure to an i.v. (instrumental variable) type estimator with the availability of
instruments. Section 6 explores the relative finite sample performance of the estimator in
two ways- section 6.1 reports results from a simulation study, and 6.2 applies the estimator
to explore a comparison of two courses of treatment for drug abuse. Section 7 concludes
by summarizing results and discussing areas for future research. A mathematical appendix
provides the details of the proofs of the asymptotic theory results.
2
Inequality Conditions in Randomly Censored Regression Models
Throughout the rest of this paper we will be concerned with inference on the k dimensional
parameter vector β0 in the model
yi = x′i β0 + ǫi
(2.1)
where xi is a k-dimensional vector of covariates, β0 is the unknown parameter of interest,
and ǫi denotes the unobserved error term. Complications arise due to the censoring of the
Khan and Tamer 5
outcome yi . In particular, we observe the random vector (xi , vi , di ) such that
vi = min(yi , ci )
di = I(yi < ci )
where vi is a scalar variable and di is a binary variable that indicates whether an observation
is censored or not. The random variable ci denotes the censoring variable that is only
observed for censored observations, and I[·] denotes an indicator function, taking the value
1 if its argument is true and 0 otherwise. In the absence of censoring, x′i β0 + ǫi would be
equal to the observed dependent variable, which in the accelerated failure time model context
will usually be the log of survival time. In the censored model, the log-survival time is only
partially observed. Another example of the above model can be a Roy/competing risk model
where y can be denoted as the negative of wage in sector 1 and c is the negative of wage in
sector 2 and one observes y for worker i in sector 1 if and only if yi > ci .
Next, we describe a set of assumptions that we use for inference on β0 . We first start
with a conditional median assumption.
A1 med(ǫi |xi ) = 0 where yi = x′i β0 + ǫi
This assumption restricts the conditional median5 of ǫ|x. Our model is thus based on quantile
restrictions on durations, which are similar to assumptions made in the quantile regression
literature- see Koenker and Bassett (1978). The median restriction here is without loss
of generality and our results apply for any quantile. In the case of censoring, our median
restriction is similar to one used in other models in the literature- e.g. Powell (1984), Honoré,
Khan, and Powell (2002) and Ying, Jung, and Wei (1995). It permits general forms of
heteroskedasticity, and is weaker than the independence assumption ǫi ⊥ xi as was imposed
in Buckley and James (1979)), Yang (1999), and Portnoy (2003). This assumption alone, in
the presence of random censoring, provides inequality restrictions on a set of appropriately
defined functions. Let the functions τ1 (xi , β) and τ0 (xi , β) be defined as:
τ1 (xi , β) = E[I[vi ≥ x′i β] xi ] −
1
2
τ0 (xi , β) = E[(1 − di ) + di I[vi > x′i β] xi ] −
1
1
= − E[di I[vi ≤ x′i β]|xi ]
2
2
We can show that the above functions, when evaluated at the true parameter, satisfy inequality restrictions. This is described in the following lemma.
5
Implicitly, this assumption requires that this median is unique and hence that the conditional distribution
of ǫ|x is strictly increasing around zero.
Khan and Tamer 6
Lemma 2.1 At the truth (β = β0 ), and on SX , the support6 of x, the following holds:
∀xi ∈ SX , τ1 (xi , β0 ) ≤ 0 ≤ τ0 (xi , β0 )
(2.2)
proof: First, we have for τ1 :
τ1 (xi , β0 ) = E[I[vi ≥ x′i β0 ] xi ] −
1
2
1
2
1
′
= E[I[ǫi ≥ 0; ci ≥ xi β0 ] xi ] −
2
1
≤ E[I[ǫ ≥ 0] xi ] − = 0
2
= E[min(yi, ci ) ≥ x′i β0 ] xi ] −
where the inequality follows from the fact that {ǫi ≥ 0; ci ≥ x′i β0 } ⊆ {ǫi ≥ 0}. Now, for τ0 :
τ0 (xi , β0 )
=
=
=
≥0
1
− E[di I[vi ≤ x′i β0 ]|xi ]
2
1
− E[I[yi ≤ ci , yi ≤ x′i β0 ]|xi ]
2
1
− E[I[ǫi ≤ ci − x′i β0 , ǫi ≤ 0]|xi ]
2
where the inequality follows using similar arguments as above.
As we can see, the randomly censored regression model can be written as a conditional
moment inequalities model. With only assumption A.1, the model provides the above
inequality restrictions that hold at the true parameter β0 . So, these inequalities can be used
to construct the set ΘI that contains observationally equivalent parameters (including β0 ).
In general and under only A.1, this set is not a singleton. However, this set is robust to any
kind of correlation between ci and xi and also ci and ǫi and so allows for general types of
censoring that can be random and can be endogenous, or dependent on yi (conditional on
xi )7 . The statistical setup is exactly one of competing risks where the risks are allowed to
6
Throughout we will be assuming that the regressors (with the exception of the constant corresponding
to the intercept term in β) will be continuously distributed on SX . This assumption is only made for
notational convenience, though as pointed out to us by a referee, in the discrete regressor case, both finite
sample performance and limiting distribution theory will be affected if strict inequalities are used in our
objective function.
Khan and Tamer 7
be dependent, or a Roy model setup. It is well known that a general dependent competing
risk model is not (point) identified- see, e.g. ?. So, additional assumptions are needed to
shrink the set to a point.
It is possible to obtain sufficient point identification conditions in some censored regression models. Heuristically, one looks for conditions under which for every b 6= β0 , we can
find a set of positive measure for x, where one of the two inequalities above is reversed. For
example, in the case of fixed (right) censoring at zero (ci = 0), the set ΘI shrinks to a point
if the set of xi ’s for which x′i β0 is negative has positive mass and xi x′i is full rank on this
set. This intuition was used to derive similar point identification conditions in Manski and
Tamer (2002) and was also essentially shown by Powell (1984). We extend this intuition
to our model setup where the censoring is allowed to be random and correlated with the
regressors and error terms.
Let β 6= β0 and let x′i δ = x′i β − x′i β0 . Then, we have
τ1 (xi , β) = P (ǫ ≥ x′i δ; ci ≥ x′i β|xi ) −
τ0 (xi , β) =
1
2
1
− P (ǫ ≤ x′i δ; ǫi ≤ ci − x′i β0 |xi )
2
(2.3)
(2.4)
So, a sufficient condition for point identification is for (2.3) to be positive for some xi , or for
(2.4) to be negative. One needs to find such xi ’s for every b 6= β0 . A sufficient condition for
this is the following assumption.
A2 The subset
C = {xi ∈ SX : Pr(ci ≥ x′i β0 |xi ) = 1}
does not lie in a proper linear subspace of Rk
A2 imposes relative support conditions on the censoring variable and the index. Specifically
it requires the regressor values for which the lower support point of the censoring distribution
(which we permit to vary with the regressor values) exceeds the index value.
7
By construction, we have y1 ≤ y ≤ y2 where y1 = y × d + (1 − d) × (−∞) and y2 = y × d + (1 − d) × c.
Hence, by A.1, we have med(y1 |x) ≤ x′ β0 ≤ med(y2 |x). Now, define the set Θ = {b : M ed(y1 |x) ≤ x′ b ≤
M ed(y2 |x)}. This set is the sharp set, since for any b ∈ Θ, there exists a y ∈ [y1 , y2 ] such that M ed(y|x) = x′ b
(of course this presumes that Θ is nonempty, since β0 belongs to it.). These bounds on medians contain
exactly the same information as the inequalities in (2.2) above and hence the identified set constructed using
(2.2) is sharp.
Khan and Tamer 8
A2 is a sufficient condition for identification and it is related to the notion of “regular
identification” since this condition is necessary (but not sufficient) for root n estimation
of β0 . For example, when (yi, xi , ci ) are jointly normal, we see that C has measure 0, but
it can be shown that β0 here is point identified. This type of “identification at infinity”
in the jointly normal model is delicate and typically leads to slower rates of convergence,
analogous to results in Andrews and Schafgans (1998). To estimate β0 at the regular rate,
we strenghten A2 in section 3 below and require that the set C has positive measure.
We note also that this condition easily allows for the fixed censoring case, and it reduces
to the condition in Powell (1984) and Honoré, Khan, and Powell (2002). Our ability to
accommodate the fixed censoring case is in contrast with other work which imposes relative
support conditions- e.g. Ying, Jung, and Wei (1995) and Koul, Susarla, and Ryzin (1981).
We also note that A2 does not impose relative support conditions on the latent error term,
ǫi . This is in contrast to the condition in Koul, Susarla, and Ryzin (1981) which requires that
the censoring variable ci exceeds the support of x′i β0 + ǫi , effectively getting identification
from values of the regressors where censoring cannot occur.
Remark 2.1 Assumptions A.1 and A.2 allows for correlation between the censoring variable, the regressors and the latent error terms. On the other hand, statistical independence
between ǫi and (xi , ci ) is imposed in Buckley and James (1979) and Yang (1999), Portnoy
(2003). Independence between ci and (xi , ǫi ) was imposed in Ying, Jung, and Wei (1995),
and Honoré, Khan, and Powell (2002) 8 . This is important in competing risks setups since
assuming independence between durations is strong. In Roy model economies, independence
is ruled out since one’s skill in one sector is naturally correlated to his/her skill in another9
sector. Our conditions above permit conditional heteroskedasticity, covariate dependent censoring, and even endogenous censoring (ci dependent on ǫi ), which is more general than
the conditional independence condition ci ⊥ ǫi |xi . The cost of these generalities is in terms
of strong point identification conditions. A.2 requires support restrictions on the censoring
variables relative to the index. Naturally, under only assumption A.1, the model identifies a
set of parameters that includes β0 .
8
Both of these papers suggest methods to allow for the censoring variable to depend on the covariates.
These methods involve replacing the Kaplan-Meier procedure they use with a conditional Kaplan Meier which
will require the choice of smoothing parameters to localize the Kaplan-Meier procedure, as well as trimming
functions and tail behavior regularity conditions. Furthermore, it is not clear how to allow endogenous
censoring in their setups.
9
In that literature, one models the process for c also, typically assuming that ci = zi′ γ0 +νi , where exclusion
restrictions (some variable in z are not in x) and support or continuity condition deliver identification of β0
(and γ0 ). See Heckman and Honoré (1990).
Khan and Tamer 9
The next lemma provides a set of inequality restrictions that only hold at β0 . Those
inequalities are in terms of the observed variables (vi , di, xi ).
Lemma 2.2 Let assumptions A1-A2 hold. For each β 6= β0
P xi ∈ SX : τ1 (xi , β) > 0 or τ0 (xi , β) < 0 > 0
(2.5)
and the parameter of interest β0 is point identified.
Lemma 2.1 states that at the true value the function τ1 is negative and τ0 is positive everywhere on the support of xi , while lemma 2.2 states that at β 6= β0 , either τ1 is strictly
positive somewhere on the support of xi or τ0 is strictly negative somewhere on the support
of xi . Hence, under A2 (and A1), the true parameter β0 is point identified.
Proof of Lemma 2.2: Recalling τ1 (xi , β) = P (ǫ ≥ x′i δ; ci ≥ x′i β|xi ) − 21 and
τ0 (xi , β) = 21 − P (ǫ ≤ x′i δ; ǫi ≤ ci − x′i β0 |xi ) consider first the case when x′i δ < 0. Then for x
satisfying condition A2 for β0 ,
x′i δ < 0 =⇒ (2.3) ≥ P (ǫi ≥ x′i δ; ci ≥ x′i β0 |xi ) −
(1)
= P (ǫi ≥ x′i δ|xi ) −
1
2
1
2
(2)
> 0
where (1) follows from x belonging to the special set postulated in A2, and (2) follows from 0
being the unique conditional median. Now for the case when x′i δ > 0. Consider xi satisfying
the condition in A2. Hence for that xi , we have
x′i δ > 0 =⇒ (2.4) < 0
where the inequality follows from the fact that x′i δ is positive and ci ≥ xi β0 .
Remark 2.2 Condition A2 is sufficient for the model to point identify the parameter, but
if A2 is considered too strong in some settings, then one can maintain only assumption
A1 and consistently estimate the set of parameters which include the truth, β0 , using the
estimator we propose below (this estimator is “adaptive” in the sense that it estimates the
identified set, which under A2 is the singleton β0 ). Another way to obtain weaker sufficient
Khan and Tamer 10
point identification conditions is require independence between ci and ǫi as in Honoré, Khan,
and Powell (2002). In some empirical settings, it is plausible to maintain independence
between ci and ǫi . These include some statistical experiments where the censoring is random
and is set independently of the process that generated the outcome (usually the censoring
is part of the experimental design). In other settings, especially in economic applications,
independence or even conditional independence can be suspect, since censoring can be affected
by unobservables that also affect outcomes and so maintaining independence would lead to
inconsistent estimates (that might not fall in the identified set).
Remark 2.3 Our identification result in Lemmas 2.1 and 2.2 uses available information
in the two functions τ1 (·, ·) and τ0 (·, ·). We can contrast this with the procedure in Ying,
Jung, and Wei (1995) which is only based on the function τ1 (·, ·), and consequently requires
reweighting the data using the Kaplan-Meier estimator. As alluded to previously, this imposes
support conditions which can rule out, among other things, fixed censoring, and does not allow
for covariate dependent censoring, unless one uses the conditional Kaplan Meier estimator of
Beran (1981). Reweighting the data by the conditional Kaplan Meier is complicated since it
involves smoothing parameters and trimming procedures, and furthermore, it does not address
the problem of endogenous censoring.
We conclude this section by pointing out that our identification results readily extend
to the doubly censored regression model. For this model the econometrician observes the
doubly censored sample, which we can express as the pair (vi , di ) where
di = I[c1i < x′i β0 + ǫi ≤ c2i ] + 2 · I[x′i β0 + ǫi ≤ c1i ] + 3 · I[c2i < x′i β0 + ǫi ]
vi = I[di = 1] · (x′i β0 + ǫi ) + I[di = 2]c1i + I[di = 3]c2i
where c1i , c2i denote left and right censoring variables, whose distributions may depend on
(xi , ǫi ) and who satisfy P (c1i < c2i ) = 1.
Note we have the following two conditional moment inequalities:
E[I[di 6= 2]I[vi ≥ x′i β0 ]|xi ] −
1
≤0
2
(2.6)
1
− E[I[di 6= 3]I[vi ≤ x′i β0 ]|xi ] ≥ 0
(2.7)
2
from which we can redefine the functions τ0 (xi , β), τ1 (xi , β) accordingly. With these functions
we can set identify the parameter vector under A1. To get point identification in the double
censoring case we would change A2 to:
Khan and Tamer 11
A2 The subset
CD = {xi ∈ SX : Pr(c1i ≤ x′i β0 ≤ c2i |xi ) = 1}
does not lie in a proper linear subspace of Rk
3
Estimation: Transforming the Conditional Moment
Inequalities Model
In this section, we propose an objective function that can be used to conduct inference on
the parameter10 β0 . We study the large sample properties of the extrema of this objective
function in the case of point identification, i.e. when A2 above holds. In cases where A2
does not hold, one can use set estimation methods such as those in Chernozhukov, Hong,
and Tamer (2002) using the same objective function.
Since our identification results are based on conditional moment inequalities holding for
all values xi , one might be tempted to use similar moments that are unconditional11 on xi .
However, in general, this strategy might yield a loss of information, i.e., the implied model
is not able to point identify β0 . So, to ensure identification, an estimator must preserve
the information contained in the conditional inequalities, i.e., ensure that the inequalities
hold for all xi , a.e. Our estimation procedure will have to differ from frameworks used to
translate a conditional moment model (based on equality constraints) into an unconditional
moment model while ensuring global identification of the parameters of interest. To avoid
estimating conditional distributions (which involve smoothing parameters), we extend insights from works by Bierens (1990), Stute (1986), Koul (2002), Chen and Fan (1999) and
recently Dominguez and Lobato (2004) and transform the conditional moment inequalities
into unconditional ones while preserving the informational content.
We first define the following functions of β, and two vectors of the same dimension as xi .
Specifically let t1 , t2 denote two vectors the same dimension as xi and let H1 (.) and H2 (.) be
10
In this and subsequent sections we will focus on the singly censored model. We note from our remark
in the previous section that we can easily modify our objective function to conduct inference in the doubly
censored model as well.
11
For example, E[m(yi ; xi )|xi ] ≤ 0 ⇒ E[w(xi )m(yi ; xi )] ≤ 0, where w(xi ) is an appropriate positive weight
function.
Khan and Tamer 12
defined as follows:
H1 (β, t1 , t2 ) = E {τ1 (xi ; β)I[t1 ≤ xi ≤ t2 ]} = E [I[vi ≥
x′i β]
1
− ]I[t1 ≤ xi ≤ t2 ] (3.1)
2
1
′
H0 (β, t1 , t2 ) = E {τ0 (xi ; β)I[t1 ≤ xi ≤ t2 ]} = E [ − di I[vi ≤ xi β]]I[t1 ≤ xi ≤ t2 ] (3.2)
2
where the above inequality t1 ≤ xi is to be taken componentwise. From fundamental properties of expectations,12 conditional moment conditions can be related to unconditional moment
conditions with indicator functions as above comparing regressor values to all vectors on the
support of x. As we will see below, this translates into estimation procedures involving a
third order U-process. The crucial point to notice about the functions H0 , H1 is that although they preserve the information contained in τ0 and τ1 above, they are not conditional
on xi . This means that our procedures will not involve estimating conditional probabilities,
and hence there will be no need for choosing smoothing parameters or employing trimming
procedures.
In general, one can transform a model with an inequality moment condition such as
E[m(yi ; β0 )|xi ] ≤ 0 for all xi
into an informationally equivalent unconditional model as
H(β0 , x1 , x2 ) = E [m(yi ; β0 )1[x1 ≤ xi ≤ x2 ]] ≤ 0 for all (x1 , x2 )
Our global identification result is based on the following objective function of distinct realizations of the observed regressors, denoted here by xj , xk :
Q(β) = Exj ,xk H1 (β, xj , xk )I[H1 (β, xj , xk ) ≥ 0] − H0 (β, xj , xk )I[H0 (β, xj , xk ) ≤ 0] (3.3)
and the following additional assumptions: first we will strengthen Assumption A2 to require
the set C to have positive measure:
A2’ The matrix E[I[xi ∈ C]xi x′i ] is of full rank.
This is a sufficient condition for regular identification of β0 . It is slightly stronger than
assumption A2 since it requires that C has positive measure. This for example rules the
12
See for example Shiryaev (1984) page 185.
Khan and Tamer 13
case in which (y, c, x) are jointly normal. Our estimation approach is still valid (provides
consistent estimates for the identified features of the model) even when A2’ (or A2) do
not hold. Next, we impose the following smoothness conditions on the error distribution,
regressors and censoring variable:
A3 The regressors xi are assumed to have compact support, denoted by X , and are continuously distributed13 with a density function fX (·) that is bounded away from 0 on
X.
A4 The error terms ǫi are absolutely continuously distributed with conditional density function fǫ (ǫ | x) given the regressors xi = x which has median equal to zero, is bounded
above, Lipschitz continuous in ǫ, and is bounded away from zero in a neighborhood of
zero, uniformly in xi .
The main identification result is stated in the following lemma:
Lemma 3.1 Under A1, Q(β) is minimized at the set ΘI , where
ΘI = {β ∈ Rk : τ1 (xi , β) ≤ 0 ≤ τ0 (xi , β) a.e. xi }
In addition, if A2’,A3,A4 hold, then Q(β) is minimized uniquely at β = β0 .
Proof: To prove the point identification result under A1,A2’,A3-A4 we first show that
Q(β0 ) = 0
(3.4)
To see why, note this follows directly from the previous lemmas which established that
τ1 (xi , β0 )I[τ1 (xi , β0 ) ≥ 0] = τ0 (xi , β0 )I[τ0 (xi , β0 ) ≤ 0] = 0 for all values of xi on its support.
Similarly, as established in that lemma, for each β 6= β0 , there exists a regressor value x∗ , such
that, under A4, for all x in a sufficiently small neighborhood of x∗ , max(τ1 (x, β)I[τ1 (x, β) ≥
0], −τ0 (x, β)I[τ0 (x, β) ≤ 0]) > 0 . Let Xδ denote this neighborhood of x∗ , which under A3
can be chosen to be contained in X . Since xi has the same support across observations, for
each x∗∗ ∈ Xδ , we can find values of xj , xk s.t. xj ≤ x∗∗ ≤ xk , establishing that Q(β) > 0. 13
This is assumption is made for convenience- discrete regressors with finite supports can be allowed for
as well.
Khan and Tamer 14
4
Consistency and Asymptotic Normality
Having shown global identification, we propose an estimation procedure, which is based
on the analogy principle, and thus minimizes the sample analog of Q(β). Our estimator
involves a third order U-process which selects the values of t1 , t2 that ensures conditioning
on all possible regressor values, and hence global identification. Specifically, we propose the
following estimation procedure:
First, define the functions:
n
X
1
b 1 (β, xj , xk ) = 1
(I[vi ≥ x′i β] − )I[xj ≤ xi ≤ xk ]
H
n i=1
2
(4.1)
n
X 1
b 0 (β, xj , xk ) = 1
H
( − di I[vi ≤ x′i β])I[xj ≤ xi ≤ xk ]
n i=1 2
(4.2)
Then, our estimator β̂ of β0 is defined as follows:
bn (β)
β̂ = arg min Q
(4.3)
β∈B
X
1
Ĥ1 (β, xj , xk )I[Ĥ1 (β, xj , xk ) ≥ 0]
= arg min
β∈B n(n − 1)
j6=k
(4.4)
−Ĥ0 (β, xj , xk )I[Ĥ0 (β, xj , xk ) ≤ 0]
where B is the parameter space to be defined below.
Remark 4.1 We note the above objective is similar to a standard LAD/median regression
objective function, since for a random variable z, we can write |z| = zI[z ≥ 0] − zI[z ≤
0]. The difference lies in the fact that our objective function “switches” from Ĥ1 (·, ·, ·) to
Ĥ0 (·, ·, ·) when moving from the positive to the negative region. Switching functions are what
permits us to allow for general forms of censoring.
Note also that the above estimation procedure minimizes a (generated) second order Uprocess. We note also that analogously to existing rank estimators, (e.g., Han (1987), Khan
and Tamer (2005)), this provides us with an estimation procedure without the need to select
smoothing parameters or trimming procedures. However, our estimator is more computationally involved than the aforementioned rank estimators, as the functions inside the double
summation have to be estimated themselves, effectively resulting in our objective function
being a third order U-process. We discuss a way to reduce this computational burden at the
end of this section.
Khan and Tamer 15
We next turn attention to the asymptotic properties of the estimator in (4.3).
We begin by establishing consistency under the following assumptions.
C1 The parameter space B is a compact subset of Rk .
C2 We have an iid sample (di , vi , x′i )′ , i = 1, . . . , n.
The following theorem establishes consistency of the estimator; its proof is left to the
appendix.
Theorem 4.1 Under Assumptions A1,A2’,A3,A4, and C1-C2,
p
β̂ → β0
For root-n consistency and asymptotic normality, our results our based on the following
additional regularity conditions:
D1 β0 is an interior point of the parameter space B.
D2 The regressors xi and censoring values {ci } satisfy
P {| ci − x′i β |≤ d} = O(d) if kβ − β0 k < η0 ,
some η0 > 0;
The following theorem establishes the root-n consistency and asymptotic normality of
our proposed minimum distance estimator. Due to its technical nature, we leave the proof
to the appendix.
Theorem 4.2 Under Assumptions A1,A2’,A3-A4, C1-C2, and D1-D2
√
n(β̂ − β0 ) ⇒ N(0, V −1 ΩV −1 )
(4.5)
where we define V and Ω are as follows. Let
C = {x : P (ci ≥ x′i β0 |xi = x) = 1}
Adopting the notation Iijk = I[xj ≤ xi ≤ xk ], define the function
Z
G(xj , xk ) = I[[xj , xk ] ⊆ C] fǫ (0|xi )xi Iijk fX (xi )dxi
(4.6)
Khan and Tamer 16
where fX (·) denotes the regressor density function. The Hessian matrix is
V = 2E[G(xj , xk )G(xj , xk )′ ]
(4.7)
Next we define the outer score term Ω. Let
δ0i = E[G(xj , xk )Iijk |xi ](I[vi ≥ x′i β0 ] − di I[vi ≤ x′i β0 ])
(4.8)
so we can define the outer score term Ω as
′
Ω = E[δ0i δ0i
]
(4.9)
To conduct inference, one can either adopt the bootstrap or consistently estimate the
variance matrix, using a “plug-in” estimator for the separate components. As is always the
case with median based estimators, smoothing parameters will be required to estimate the
error conditional density function, making the bootstrap a more desirable approach.
We conclude this section by commenting on computational issues. The above estimator
involves optimizing a third order U-statistic. Significant computational time can be reduced
if one uses a “split sample” approach (see Honoré and Powell (1994) for an example). This
would result in an estimator that minimizes a second order U-process that is much simpler
computationally, though less efficient14 . For the problem at hand, the split sample version
of our proposed estimator would minimize an objective function of the form:
β̂SS = arg min Q̂SSn (β)
β∈B
n 1X
= arg min
Ĥ1 (β, xj , xn−j+1)I[Ĥ1 (β, xj , xn−j+1) ≥ 0]
β∈B n
j=1
− Ĥ0 (β, xj , xn−j+1 )I[Ĥ0 (β, xj , xn−j+1 ) ≤ 0]
5
(4.10)
(4.11)
Endogenous Regressors and Instrumental Variables
In this section we illustrate how the estimation procedure detailed in the previous section
can be modified to permit consistent estimation of β0 when the regressors xi as well as the
14
An alternative procedure would be a “resampling” based estimator, where one resamples O(n) regressor
values (with replacement) in the construction of the functions Ĥ1 (β, ·, ·), Ĥ0 (β, ·, ·). We leave the asymptotic
properties of this procedure for future work.
Khan and Tamer 17
censoring variable ci are endogenous. Our identification strategy now requires one to have a
vector of instrumental variables zi .
The semiparametric literature has seen recent developments in estimating censored regression models when the regressors are endogenous. For fixed censoring models, instrumental variable approaches have been proposed in Hong and Tamer (2003) and Lewbel (2000),
whereas a control function approach has been proposed in ?, Blundell and Powell (2004),
?, for nonlinear models including censored regression. However, these estimators are not
applicable in the random censoring case, even in the case when the censoring variable is
distributed independently of the covariates. Furthermore, they all require the selection of
multiple smoothing parameters and trimming procedures.
Here we propose an estimator for the randomly censored regression model with endogenous regressors. This problem arises in a variety of settings in duration analysis. For
example, in labor economics, if the dependent variable is unemployment spell length, an
explanatory variable such as the amount of training received while unemployed could clearly
be endogenous. Another example, studied more often in the biostatistics literature, is when
the dependent variable is time to relapse for drug abuse and the endogenous explanatory
variable is a course of treatment.
To estimate β0 in this setting we assume the availability of a vector of instrumental
variables zi which will be defined through the following assumptions. Here, our sufficient
conditions for identification are:
IV1 median(ǫi |zi ) = 0.
IV2 The subset of the support of instruments
CZ0 = {zi : P (x′i β0 ≤ ci |zi ) = 1}
has positive measure. Furthermore, for each δ 6= 0, at least one of the following subsets
of CZ0 also has positive measure:
CZ0− = {zi ∈ CZ0 : P (x′i δ ≤ 0|zi ) = 1}
CZ0+ = {zi ∈ CZ0 : P (x′i δ ≥ 0|zi ) = 1}
Remark 5.1 Before outlining an estimation procedure, we comment on the meaning of the
above conditions.
1. Condition IV1 is analogous to the usual condition of the instruments being uncorrelated
with the error terms.
Khan and Tamer 18
2. Condition IV2 details the relationship between the instruments and the regressors. It
is most easily satisfied when the exogenous variable(s) have support that are relatively
large when compared to the support of the endogenous variable(s).15 Empirical settings where this support condition arises is in the treatment effect literature, where the
endogenous variable is a binary treatment variable, or a binary compliance variable.
In the latter case an example of an instrumental variable is also a binary variable
indicating treatment assignment if it is done so randomly- see for example Bijwaard
and Ridder (2005) who explore the effects of selective compliance to re-employment
experiments on unemployment duration.
Our IV limiting objective function is of the form:
QIV (β) = E [H1∗ (β, zj , zk )I[H1∗ (β, zj , zk ) ≥ 0] − H0∗ (β, zj , zk )I[H0∗ (β, zj , zk ) ≤ 0]]
(5.1)
where here
1
H1∗ (β, t1 , t2 ) = E[(I[vi ≥ x′i β] − )I[t1 ≤ zi ≤ t2 ]]
2
1
H0∗ (β, t1 , t2 ) = E[( − di I[vi ≤ x′i β])I[t1 ≤ zi ≤ t2 ]]
2
(5.2)
(5.3)
And our proposed IV estimator minimizes the sample analog of QIV (β), using
n
1
1X
(I[vi ≥ x′i β] − )I[zj ≤ zi ≤ zk ]
Ĥ1 (β, zj , zk ) =
n i=1
2
(5.4)
n
1X 1
Ĥ0 (β, zj , zk ) =
( − di I[vi ≤ x′i β])I[zj ≤ zi ≤ zk ]
n i=1 2
(5.5)
The asymptotic properties of this estimator are based on regularity conditions analogous
to those in the previous section. The theorem below establishes the limiting distribution theory for this estimator. Its proof is omitted since it follows from virtually identical arguments
used to prove the previous theorem.
15
Lewbel(2000) also requires an exogenous variable with relatively large support. However, his conditions
are stronger than those imposed here in the sense that he assumes the exogenous variable has to have
large support when compared to both the endogenous variable(s) and the error term, effectively relying on
identification at infinity. In the censored setting, his assumption corresponds to obtaining identification from
the region of exogenous variable(s) space where the data is uncensored with probability 1.
Khan and Tamer 19
Theorem 5.1
√
−1
−1
n(β̂IV − β0 ) ⇒ N(0, VIV
ΩIV VIV
)
(5.6)
where we define VIV and ΩIV are as follows. Let
C = {z : P (ci ≥ x′i β0 |zi = z) = 1}
Adopting the notation Iijk = I[zj ≤ zi ≤ zk ], define the function
Z
G(zj , zk ) = I[[zj , zk ] ⊆ C] fǫ (0|zi )E[xi |zi ]Iijk fZ (zi )dzi
(5.7)
where fZ (·) denotes the instrument density function. The Hessian matrix is
VIV = 2E[G(zj , zk )G(zj , zk )′ ]
(5.8)
Next we define the outer score term ΩIV . Let
δ0i = E[G(zj , zk )Iijk |zi ](I[vi ≥ x′i β0 ] − diI[vi ≤ x′i β0 ])
(5.9)
so we can define the outer score term Ω as
′
]
ΩIV = E[δ0i δ0i
6
(5.10)
Finite Sample Performance
The theoretical results of the previous section give conditions under which the randomlycensored regression quantile estimator will be well-behaved in large samples. In this section,
we investigate the small-sample performance of this estimator in two ways, first by reporting
results of a small-scale Monte Carlo study, and then considering an empirical illustration.
Specifically, we first study the effects of two courses of treatment for drug abuse on the time
to relapse ignoring potential endogeneity, and then we control for selective compliance to
treatment using our proposed i.v. estimator.
6.1
Simulation Results
The model used in this simulation study is
yi = min{α0 + x1i β0 + x2i γ0 + ǫi , ci }
(6.1)
Khan and Tamer 20
where the regressors x1i , x2i were chi-squared, 1 degree of freedom, and standard normal
respectively. The true values α0 , β0 , γ0 of the parameters are -0.5, -1, and 1, respectively. We
considered two types of censoring- covariate independent censoring, where ci was distributed
standard normal, and covariate dependent censoring, where we set ci = −x21i − x2i .16
We assumed the error distribution of ǫi was standard normal. In addition, we simulated
designs with heteroskedastic errors as well: ǫi = σ(x2i ) · ηi, with ηi having a standard normal
distribution and σ(x2i ) = exp(0.5 ∗ x2i ). We also simulated a design with endogenous censoring where there was one regressor, distributed standard normal, the error was homoskedastic,
and the censoring variable was related to ηi as ci = (4 + ui) ∗ ηi2 where ui was an independent
random variable distributed uniformly on the unit interval.
For these designs, the overall censoring probabilities vary between 40% and 50%. For
each replication of the model, the following estimators were calculated17 :
a) The minimum distance least absolute deviations (MD) estimator introduced in this paper.
b) The randomly censored LAD introduced in Honoré, Khan, and Powell (2002), referred
to as HKP.
c) The estimator proposed by Buckley and James (1979);
d) The estimator proposed by Ying, Jung, and Wei (1995), referred to as YJW;
Both YJW and MD estimators were computed using the Nelder Meade simplex algorithm.18 The randomly-censored least absolute deviations estimator (HKP) was computed
using the iterative Barrodale-Roberts algorithm described by Buchinsky(1995)19 ; in the random censoring setting, the objective function can be transformed into a weighted version of
the objective function for the censored quantile estimator with fixed censoring.
The results of 401 replications of these estimators for each design, with sample sizes of
50, 100, 200, and 400, are summarized in Tables I-V, which report the mean bias, median
bias, root-mean-squared error, and mean absolute error. These 5 tables corresponded to
16
We note that for this design our set C defined in Assumption A2 does not have positive measure, violating
the sufficient condition for regular identification of our estimator. Nonetheless, as we see in the simulation
tables our estimator performs relatively well in finite samples.
17
The simulation study was performed in GAUSS and C++. Codes for the estimators introduced in this
paper are available from the authors upon request.
18
OLS, LAD, and true parameter values were used in constructing the initial simplex for the results
reported.
19
OLS was used as the starting value when implementing this algorithm for the simulation study.
Khan and Tamer 21
designs with 1)homoskedastic errors and covariate independent censoring, 2)heteroskedastic
errors and covariate independent censoring, 3) homoskedastic errors and covariate dependent censoring, 4) heteroskedastic errors and covariate dependent censoring, 5) endogenous
censoring. Theoretically, only the MD estimator introduced here is consistent in all designs,
and the only estimator which is consistent in designs 4 and 5,.
HKP and YJW estimators are consistent under designs 1 and 2, whereas the BuckleyJames estimator is inconsistent when the errors are heteroskedastic as is the case in designs
2 and 4.
The results indicate that the estimation method proposed here performs relatively well.
For some designs the MD estimator exhibits large values of RMSE for 50 observations, but
otherwise appears to be converging at the root−n rate.
As might be expected, the MD estimator, which does not impose homoskedasticity of the
error terms, is superior to Buckley-James when the errors are heteroskedastic. It generally
outperforms HKP and YJW estimator when the censoring variable depends on the covariates.
This is especially the case when the errors are heteroskedastic, as the proposed estimator is
the only estimator which performs reasonably well. Table 5 indicates that MD is the only
consistent estimator for the endogenous censoring design, as it is the only estimator whose
values of bias and RMSE decline with sample size.
6.2
Empirical Example: Drug Relapse Duration
We apply the minimum distance procedure to the drug relapse data set used in Hosmer and
Lemeshow (1999), who study the effects of various variables, on time to relapse. Those not
relapsing before the end of the study are regarded as censored. Similar data were used in
Portnoy (2003).
The data is from the University of Massachusetts Aids Research Unit Impact Study.
Specifically, the data set is from a 5-year (1989-1994) period comprising of two concurrent
randomized trials of residential treatment for drug abuse. The purpose of the original study
was to compare treatment programs of different planned durations designed to prevent drug
abuse and to also determine whether alternative residential treatment approaches are variable
in effectiveness. One of the sites, referred to here as site A randomly assigned participants
to 3- and 6-month modified therapeutic communities which incorporated elements of health
education and relapse prevention. Here clients were taught how to recognize “high-risk”
situations that are triggers to relapse, and taught the skills to enable them to cope with these
Khan and Tamer 22
situations without using drugs. In the other site, referred to here as site B, participants were
randomized to either a 6-month or 12-month therapeutic community program involving a
highly structured life-style in a communal living setting. This data set contains complete
records of 575 subjects.
Here, we use the log of relapse time as our dependent variable, and the following six
independent variables: SITE(drug treatment site B=1, A=0), IV(an indicator variable taking the value 1 if subject had recent IV drug use at admission), NDT(number of previous
drug abuse treatments), RACE(white(0) or “other”(1)), TREAT (randomly assigned type
of treatment, 6 months(1) or 3 months(0)), FRAC (a proxy for compliance, defined as the
fraction of length of stay in treatment over length of assigned treatment). Table VI reports results for the 4 estimators used in the simulation study as well as estimators of two
parametric models- the Weibull and Log-Logistic. Standard errors are in parentheses.
Qualitatively, all estimators deliver similar results in the sense that the signs of the
coefficients are the same. However there are noticeable differences in the values of these
estimates, as well as their significance20 . For example, the Weibull estimates are noticeably different from all other estimates, including the other parametric estimator, in most
categories, showing a larger (in magnitude) IV effect, which is statistically insignificant for
many of the semiparametric estimates, and a smaller TREAT effect. The semiparametric
estimators differ both from the parametric estimators as well as each other. The proposed
minimum distance estimator, consistent under the most general specifications compared to
the others, yields a noticeably smaller (in magnitude) SITE effect, and with the exception
of the HKP estimator, a larger TREAT effect.
We extend our empirical study by applying the IV extension of the proposed minimum
distance estimator to the same data set. The explanatory variable length of stay (LOS) could
clearly be endogenous because of “selective compliance”. Specifically, those who comply
more with treatment (i.e. have larger values of LOS) may not be representative of the
people assigned treatment, in which case the effect of an extra day of treatment would
be overstated by estimation procedures which do not control for this form of endogeneity.
Given the random assignment of the type of treatment, the treatment indicator (TREAT) is
a natural choice (see, e.g. Bloom (1984)) of an instrumental variable as it is correlated with
LOS.
We consider estimating a similar model to one considered above, now modelling the
relationship between the log of relapse time and the explanatory variables IV, RACE, NDT,
20
Reported standard errors for the 4 semiparametric estimators were obtained from the bootstrap, using
575 samples (obtained with replacement) of 575 observations.
Khan and Tamer 23
SITE, and LOS. Table VII reports results from 4 estimation procedures: 1) ordinary least
squares (OLS) 2) two stage least squares (2SLS) using TREAT as an instrument for LOS
3) our proposed minimum distance estimator (MD) and 4) our proposed extension to allow
for endogeneity (MDIV) using TREAT as an instrument for LOS. We note that OLS and
2SLS are not able to take into account the random censoring in our data set. Nonetheless the
results from OLS and 2SLS are similar to MD and MDIV respectively. Most importantly both
procedures do indeed indicate selective compliance to treatment as the estimated coefficient
on LOS is larger for OLS and MD that it is for 2SLS and MDIV.
7
Conclusions
This paper introduces a new estimation procedure for models that are based on moment
inequality conditions. The procedure is applied to estimate parameters for an AFT model
with a very general censoring relationship when compared to existing estimators in the
literature. The procedure minimized a third order U-process, and did not required the
estimation of the censoring variable distribution, nor did it require nonparametric methods
and the selection of smoothing parameters and trimming procedures. The estimator was
shown to have desirable asymptotic properties and both a simulation study and application
using drug relapse data indicated adequate finite sample performance.
The results established in this paper suggest areas for future research. Specifically, the
semiparametric efficiency bound for this general censoring model has yet to be derived, and
it would be interesting to see how are MD estimator can be modified to attain the bound
in this specific model, as well as other models based on moment inequality conditions. We
leave these possible extensions for future research.
Khan and Tamer 24
References
Andrews, D., S. Berry, and P. Jia (2003): “On Placing Bounds on Parameters of Entry
Games in the Presence of Multiple Equilibria,” Working Paper.
Andrews, D., and G. Soares (2007): “Inference for Parameters Defined by Moment
Inequalities using Generalize Moment Selection,” Working Paper, Yale University.
Andrews, D. W. K., and M. Schafgans (1998): “Semiparametric Estimation of the
Intercept of a Sample Selection Model,” Review of Economic Studies, 65, 487–517.
Beran, R. (1981): “Nonparametric Regeression with Randomly Censored Survival Data,”
Technical Report, California - Berkeley.
Bierens, H. (1990): “A Consistent Conditional Moment Test of Functional Form,” Econometrica, 58, 1443–1458.
Bijwaard, G., and G. Ridder (2005): “Correcting for Selective Compliance in a Reemployment Bonus Experiment,” Working Paper, forthcoming in Journal of Econometrics.
Bloom, H. (1984): “Accounting for No-shows in Experimental Evaluation Designs,” Evaluation Review, 8, 225–246.
Blundell, R., and J. Powell (2004): “Endogeneity in Binary Response Models,” Review
of Economic Studies, 73.
Buchinsky, M., and J. Hahn (2001): “An Alternative Estimator for the Censored Quantile Regression Model,” Econometrica, 66, 653–671.
Buckley, J., and I. James (1979): “Linear Regression with Censored Data,” Biometrika,
66, 429–436.
Canay, I. (2007): “EL Inference for Partially Identified Models: Large Deviations Optimality and Bootstrap Validity,” Wisconsin Working Paper.
Chen, X., and Y. Fan (1999): “Consistent Hypothesis Testing in Semiparametric and
Nonparametric Models for Econometric Time Series,” Journal of Econometrics, 91(2),
373–401.
Chernozhukov, V., H. Hong, and E. Tamer (2002): “Inference in Incomplete Econometric Models,” Department of Economics, Princeton University.
Cosslett, S. (2004): “Efficient Semiparametric Estimation of Censored and Truncated
Regressions via a Smoothed Self-Consistency Equation,” Econometrica, 72, 1277–1293.
Dominguez, M., and I. Lobato (2004): “Consitent Estimation of Models Defined by
Conditional Moment Restrictions,” Econometrica, 72, 1601–1615.
Han, A. (1987): “The The Maximum Rank Correlation Estimator,” Journal of Econometrics, 303-316.
Khan and Tamer 25
Heckman, J., and B. Honoré (1990): “The Empirical Content of the Roy Model,”
Econometrica, 58, 1121–1149.
Hong, H., and E. Tamer (2003): “Inference in Censored Models with Endogenous Regressors,” Econometrica, 71(3), 905–932.
Honoré, B., S. Khan, and J. Powell (2002): “Quantile Regression Under Random
Censoring,” Journal of Econometrics, 64, 241–278.
Honoré, B., and J. Powell (1994): “Pairwise Difference Estimators of Censored and
Truncated Regression Models,” Journal of Econometrics, 64, 241–278.
Hosmer, D., and S. Lemeshow (1999): Survival Analysis: Regression Modelling of Time
to Event Data. Wiley, New York, NY.
Khan, S., and E. Tamer (2005): “Partial Rank Estimation of Transformation Models
with General forms of Censoring,” Working Paper, forthcoming Journal of Econometrics.
Koenker, R., and G. Bassett (1978): “Regression Quantiles,” Econometrica, 46, 33–50.
Koul, H. (2002): Weighted Empirical Processes Dynamic Nonlinear Models. SpringerVerlag, New York, NY.
Koul, H., V. Susarla, and J. V. Ryzin (1981): “Regression Analysis with Randomly
Right Censored Data,” Annals of Statistics, 9, 1276–1288.
Lewbel, A. (2000): “Semiparametric Qualitative Response Model Estimation with Unknown Heteroscedasticity or Instrumental Variables,” Journal of Econometrics, 97, 145–
177.
Manski, C., and E. Tamer (2002): “Inference on Regressions with Interval Data on a
Regressor or Outcome,” Econometrica, 70, 519–547.
Pakes, A., and D. Pollard (1989): “Simulation and the Asymptotics of Optimization
Estimators,” Econometrica, 57(5), 1027–1057.
Pakes, A., J. Porter, K. Ho, and J. Ishii (2005): “Moment Inequalities and Their
Applications,” Working Paper.
Portnoy, S. (2003): “Censored Regression Quantiles,” Journal of American Statistical
Association, 98, 1001–1012.
Powell, J. (1984): “Least Absolute Deviations Estimation for the Censored Regression
Model,” Journal of Econometrics, pp. 303–325.
Rosen, A. (2006): “Confidence Sets for Partially Identified Parameters that Satisfy a Finite
Number of Moment Inequalities,” UCL Working Paper.
Serfling, R. (1980): Approximation Theorems of Mathematical Statistics. Wiley, New
York, NY.
Khan and Tamer 26
Sherman, R. (1993): “The Limiting Distribution of the Maximum Rank Correlation Estimator,” Econometrica, 61, 123–137.
(1994a): “Maximal Inequalities for Degenerate U-Processes with Applications to
Optimization Estimators,” Annals of Statistics, 22, 439–459.
(1994b): “U-Processes in the Analysis of a Generalized Semiparametric Regression
Estimator,” Econometric Theory, 10, 372–395.
Shiryaev, A. (1984): Probbaility. Springer-Verlag, New York, NY.
Stute, W. (1986): “On Almost Sure Convergence of Conditional Empirical Distribution
Functions,” Annals of Statistics, 14(3), 891–901.
Yang, S. (1999): “Censored Median Regression Using Weighted Empirical Survival and
Hazard Functions,” Journal of the American Statistical Association, 94, 137–145.
Ying, Z., S. Jung, and L. Wei (1995): “Survival Analysis with Median Regression
Models,” Journal of the American Statistical Association, 90, 178–184.
Khan and Tamer 27
A
Proof of Theorem 4.1
Heuristically, to establish consistency, we need identification, compactness, continuity and
uniform convergence (See for example Theorem 2.1 in Newey and McFadden(1994)). Identification follows from Lemma 3.1. Compactness follows from Assumptions C1 and continuity
of the objection function follows from the smoothness conditions in A3,A4 respectively. It
remains to show uniform convergence of the sample objective function to Q(·). To establish
this result we will define the following functions to ease notation, we will show that
sup
β∈B
X
1
Ĥ1 (β, xj , xk )I[Ĥ1 (β, xj , xk ) ≥ 0] − Q1 (β) = op (1)
n(n − 1) i6=j
(A.1)
where
Q1 (β) = E[H1 (β, xj , xk )I[H1 (β, xj , xk ) ≥ 0]]
(A.2)
noting that identical arguments can be used for the component of the objective function
involving Ĥ0 (β, xj , xk ). To show (A.1) we will first show that
sup
β∈B
X
1
Ĥ1 (β, xj , xk )I[Ĥ1 (β, xj , xk ) ≥ 0] − H1 (β, xj , xk )I[H1 (β, xj , xk ) ≥ 0] (A.3)
n(n − 1) i6=j
is op (1). To show (A.3) is op (1), we first replace I[Ĥ1 (β, xj , xk ) ≥ 0] with I[H1 (β, xj , xk ) ≥ 0].
We will next attempt to show that
sup
β∈B
X
1
(Ĥ1 (β, xj , xk ) − H1 (β, xj , xk ))I[H1 (β, xj , xk ) ≥ 0] = op (1)
n(n − 1)
(A.4)
i6=j
To do so, we expand Ĥ1 (β, xj , xk ), which involved a summation of observations denoted by
subscript i. The term
X
1
1
((I[vi ≥ x′i β] − )I[xj ≤ xi ≤ xk ] − H1 (β, xj , xk ))I[H1 (β, xj , xk ) ≥ 0]
n(n − 1)(n − 2) i6=j6=k
2
is a mean 0 third order U-process. Consequently, by the i.i.d assumption in C2, and applying
Corollary 7 in Sherman(1994a), this term is uniformly op (1). It remains to show that replacing Ĥ1 (β, xj , xk ) with H1 (β, xj , xk ) inside the indicator function yields an asymptotically
uniformly negligible remainder term. Specifically, we will show that
sup
β∈B
X
1
Ĥ1 (β, xj , xk )(I[Ĥ1 (β, xj , xk ) ≥ 0] − I[H1 (β, xj , xk ) ≥ 0]) = op (1) (A.5)
n(n − 1) i6=j
Khan and Tamer 28
Noting that Ĥ1 (β, xj , xk ) is uniformly bounded in β, xj , xk , it will suffice to show that
sup
β∈B
X
1
|I[Ĥ1 (β, xj , xk ) ≥ 0] − I[H1 (β, xj , xk ) ≥ 0]| = op (1)
n(n − 1) i6=j
(A.6)
To do so, we will, w.l.o.g., only show that
P
1
i6=j I[Ĥ1 (β, xj , xk ) ≥ 0]I[H1 (β, xj , xk ) < 0] is op (1) uniformly in β. To do so owe
n(n−1)
will add and subtract the expectation of the term inside the double summation, conditional
on xj , xk . We note that by subtracting this conditional expectation from the double summation, we again have a mean 0 U-process, to which we can again apply Corollary 7 in
Sherman(1994a) to conclude this is uniformly op (1).
It thus remains to show that
X
1
E[I[Ĥ1 (β, xj , xk ) ≥ 0]I[H1 (β, xj , xk ) < 0]|xj , xk ]
n(n − 1) i6=j
is op (1) uniformly in β.
We condition on values of xj , xk and decompose I[H(β, xj , xk ) < 0] as
I[H1 (β, xj , xk ) < 0] = I[H1 (β, xj , xk ) ≤ −δn ] + I[H1 (β, xj , xk ) ∈ (−δn , 0)]
where δn =
1
.
log n
(A.7)
We will first focus on the term
I[Ĥ1 (β, xj , xk ) ≥ 0]I[H1 (β, xj , xk ) ≤ −δn ]
(A.8)
The probability (i.e. conditional expectation) that the above product of indicators is positive,
recalling that we are conditioning on xj , xk , is less than or equal to the probability that
n
X
1
(I[vi ≥ x′i β] − )Iijk − H1 (β, xj , xk ) > −nH1 (β, xj , xk )
2
i=1
(A.9)
with H1 (β, xj , xk ) ≤ −δn , where above Iijk = I[xj ≤ xi ≤ xk ]. The conditional probability
of the above event is less than or equal to the conditional probability that
n
X
1
(I[vi ≥ x′i β] − )Iijk − H1 (β, xj , xk ) > nδn
2
i=1
(A.10)
to which we can apply Hoeffding’s inequality, see, e.g. Pollard(1984), to bound above by
exp(−2nδn2 ).
Khan and Tamer 29
Note this bound is independent of β, xj , xk , and converges to 0 at the rate n−2 , establishing the uniform (across β, xj , xk ) convergence of E[I[Ĥ(β, xj , xk ) ≥ 0]I[H(β, xj , xk ) ≤
−δn ]|xj , xk ] to 0. We next show the uniform convergence of
X
1
I[Ĥ1 (β, xj , xk ) ≥ 0]I[H1 (β, xj , xk ) ∈ (−δn , 0)]
n(n − 1) i6=j
(A.11)
for which it will suffice to establish the uniform convergence of:
X
1
I[H1 (β, xj , xk ) ∈ (−δn , 0)]
n(n − 1)
(A.12)
i6=j
to 0. Subtracting E[I[H1 (β, xj , xk ) ∈ (−δn , 0)]] from the above summation we can again
apply Corollary 7 in Sherman(1994a) to conclude that this term is uniformly in β op (1).
The expectation E[I[H(β, xj , xk ) ∈ (−δn , 0)]] is uniformly op (1) by applying the dominated
convergence theorem. Combining all our results we conclude that (A.3) is op (1).
Next, we will establish that
sup
β∈B
X
1
H1 (β, xj , xk )I[H1 (β, xj , xk ) ≥ 0] − Q1 (β) = op (1)
n(n − 1) i6=j
(A.13)
For this we can apply existing uniform laws of large numbers for centered U− processes.
Specifically, we can show the r.h.s. of (A.13) is Op (n−1/2 ) by Corollary 7 in Sherman(1994a)
since the functional space index by β is Euclidean for a constant envelope. The Euclidean
property follows from example (2.11) in Pakes and Pollard (1989).
B
Proof of Theorem 4.2
Having shown consistency, our proof strategy will be to approximate the objective function
bn (.), locally in a neighborhood of β0 , by a an appropriate quadratic in β function. The
Q
approximation needs to allow for the fact that this objective function is not smooth in β.
Quadratic approximation of objective functions have been provided in, for example, Pakes
and Pollard (1989), and Sherman (1994a), (1994b),(1993) among others. First we establish
root-n consistency. For root-n consistency we will apply Theorem 1 of Sherman (1994b).
Keeping our notation deliberately close to Sherman(1994b), here we denote our sample
objective function Q̂n (β) by Gn (β) and denote our limiting objective function Q(β) by G(β).
From Theorem 1 in Sherman(1994b), sufficient conditions for rates of convergence are that
1. β̂ − β0 = Op (δn )
Khan and Tamer 30
2. There exists a neighborhood of β0 and a constant κ > 0 such that G(β) − G(β0 ) ≥
κkβ − β0 k2 for all β in this neighborhood.
3. Uniformly over Op (δn ) neighborhoods of β0
√
Gn (β) = G(β) + Op (kβ − β0 k/ n) + op (kβ − β0 k2 ) + Op (ǫn )
(B.1)
1/2
which suffices for β̂ − β0 = Op (max(ǫn , n−1/2 )). With this theorem we will establish root-n
consistency in two stages. Having already established consistency we will first set δn = o(1),
ǫn = O(n−1/2 ), and show the above three conditions are satisfied. This will imply that the
estimator is fourth-root consistent. We will then show the conditions of the theorem are
satisfied for δn = O(n−1/4 ) and ǫn = O(n−1 ), establishing root-n consistency.
Regarding the showing the first result of fourth root consistency, in what follows throughout the rest of our proof, it will prove convenient to subtract the following term, which does
not depend on β, from our objective function:
X
1
Ĥ1 (β0 , xj , xk )I[H1 (β0 , xj , xk ) ≥ 0] − Ĥ0 (β0 , xj , xk )I[H0 (β0 , xj , xk ) ≤ 0] (B.2)
n(n − 1) j6=k
We note that since β does not enter (B.2), the value of the estimator is not affected by
including this additional term. We also note that the expectation of this term conditional
on xj , xk is 0.
To show the second of the three conditions, we will first derive an expansion for G(β)
around G(β0 ). We note that even though Gn (β) is not differentiable in β, G(β) is sufficiently
smooth for Taylor expansions to apply as the expectation operator is a smoothing operator
and the smoothness conditions in Assumption A3,A4,D2 imply differentiability after taking
expectations. Taking a second order expansion of G(β) around G(β0 ), we obtain
1
G(β) = G(β0 ) + ∇β G(β0 )′ (β − β0 ) + (β − β0 )′ ∇ββ G(β ∗ )(β − β0 )
2
(B.3)
where ∇β and ∇ββ denote first and second derivative operators and β ∗ denotes an intermediate value. We note that the first two terms of the right hand side of the above equation
are 0, the first by how we defined the objective function, and the second by our identification
result in Theorem 1. We will later formally show that
∇ββ G(β0 ) = V
(B.4)
and V is invertible by Assumption A2’, so we have
(β − β0 )′ ∇ββ G(β0 )(β − β0 ) > 0
(B.5)
Khan and Tamer 31
∇ββ G(β) is also continuous at β = β0 by Assumptions A3 and D2, so there exists a neighborhood of β0 such that for all β in this neighborhood, we have
(β − β0 )′ ∇ββ G(β)(β − β0 ) > 0
(B.6)
which suffices for the second condition to hold.
To show the third condition, our first step is to replace the indicator functions in the objective function, I[Ĥ1 (β, xj , xk ) ≥ 0], I[Ĥ0 (β, xj , xk ) ≤ 0], with the functions I[H1 (β, xj , xk ) ≥
0], I[H0 (β, xj , xk ) ≤ 0] respectively, and derive the corresponding representation. (We will
deal with the resulting remainder term from this replacement shortly) We expand the terms
Ĥ1 (β, xj , xk ) and Ĥ0 (β, xj , xk ), first, exclusively dealing with the first expansion since the
second is similar. This results in the third order U-process:
X
X
1
1
1
I[H1 (β, xj , xk ) ≥ 0](I[vl ≥ x′l β] − )Iljk ≡
m(zj , zk , zl()B.7)
n(n − 1)(n − 2)
2
n(n − 1)(n − 2)
j6=k6=l
j6=k6=l
where zi = (xi , vi ), Iljk = I[xj ≤ xl ≤ xk ]. (B.7) is a third order U-statistic and we analyze
its properties by representing it as a projection plus a degenerate U-process- see, e.g. Serfling (1980). Note that the unconditional expectation corresponds to the first ”half” of the
limiting objective function, which recall here we denoted by G(β). We will evaluate representations for expectations conditional on each of the three arguments, minus the unconditional
expectation. In particular, first we write (B.7) as:
(B.7) = Pn P m(zj , ., .) + Pn P m(., zk , .) + Pn P (., ., zl ) + Un h
(B.8)
P
where Pn P m(zj , ., .) is equal to n1 j Pk,l m(zj , zk , zl ) where Pk,l denotes expectation with
respect to second and third arguments, and Un h is a degenerate U-statistic (see Sherman
(1994a)). Hence, to get the first order term in (B.1), we need to take the first order expansion
of the projection terms in (B.8).
We first turn attention to the expectation conditional on the third argument l. This
summation will be of the form
n
1
1X
(I[vl ≥ x′l β] − )E [I[H1 (β, xj , xk ) ≥ 0]Iljk | xl ]
(B.9)
n l=1
2
To get the Op ( √1n ) term in (B.1), we will take a mean value expansion of the term inside the
above summation around β = β0 . Note by our normalization (i.e the subtraction of (B.2)
from the objective function), the initial replacement of β with β0 yields a term that is op ( n1 )
uniformly in β in op (1) neighborhoods of β0 . Notice also that I[H1 (β0 , xj , xk ) ≥ 0] · Iljk = 1
implies that [xj , xk ] ⊂ C, as is xl ; we next evaluate
∇β E [I[H1 (β, xj , xk ) ≥ 0]Iljk |xl ]
(B.10)
Khan and Tamer 32
at β = β0 . Here by definition of H1 , we have:
Z
1
′
′
fX (u)du (B.11)
H1 (β, xj , xk ) = I[xj ≤ u ≤ xk ] P c ≥ u β; ǫ ≥ u (β − β0 )|u −
2
When integrating over the different values of xj , xk , we decompose the set of values into
those satisfying the interval [xj , xk ] contained in C and those that do not. We do this
because H1 (β0 , xj , xk ) < 0 in the latter case and the expectation term in (B.10) is 0. Two
applications of the dominated convergence theorem on the subset of values where [xj , xk ] is
contained in C yields that (B.10) is of the form:
Z
E [G(xj , xk )] = E I[[xj , xk ] ⊆ C] fǫ|X (0|u)Iujk ufX (u)du
This is so since ∇β P c ≥ u′ β; ǫ ≥ u′ (β − β0 )|u = −fǫ|x (0) on the set C. So combining this
expansion term with (B.9), yields
n
1
1X
I[xl ∈ C](I[vl ≥ x′l β0 ] − )E [G(xj , xk )]′ (β − β0 )
n l=1
2
Next we evaluate
n
1X
1
∇β (I[vl ≥ x′l β] − )E [I[H1 (β0 , xj , xk ) ≥ 0]Iljk | xl ]
n l=1
2
(B.12)
(B.13)
This term will cancel out with the corresponding derivative term from the H0 (·, ·, ·) “half”
of the objective function, completing the representation of the linear term in our expansion,
which remains as it is in (B.12). We note that using similar arguments, along with the
law of large numbers, it follows that the remainder term in the mean value expansion of
(I[vl ≥ x′l β]− 21 )E[I[H1 (β, xj , xk ) ≥ 0]] yields a term that is op (kβ −β0 k2 ). Also, we note that
the expectation of this term, which must be subtracted from our U−statistic decomposition
can be shown to be negligible using similar arguments.
As far as the other two projections in (B.8), it can be shown that the expectation conditional
j, when combined with the corresponding conditional expectation of the other “half” of the
objective function, has the representation:
op (kβ − β0 k2 )
(B.14)
as does the expectation conditional on the second argument, indexed by k. So, the Op (||β −
√
β0 ||/ n) term is (B.1) is (B.12) and since it has expectation (across vl , xl ) of 0, and finite
variance it is bounded in probability by a CLT. So we have established that so far the sample
objective function can be represented as the limiting objective function plus
√
√
1
Op (kβ − β0 k/ n) + op (kβ − β0 k/ n) + op (kβ − β0 k2 ) + op ( )
n
(B.15)
Khan and Tamer 33
Finally, we note the higher order terms in the projection theorem are op (n−1 ) uniformly
for β in op (1) neighborhoods of β0 using arguments similar to those in Theorem 3 in Sherman(1993). So by Theorem 1 in Sherman(1994b), we are able to show, that it can be
expressed as the limiting first ”half” of the objective function plus a remainder term that is
(uniformly in op (1) neighborhoods of β0 ),
√
1
Op (kβ − β0 k/ n) + op (kβ − β0 k2 ) + op ( )
n
(B.16)
Collecting terms, we conclude that the infeasible estimator, which replaced I[Ĥ1 (β, xj , xk ) ≥
0], I[Ĥ0 (β, xj , xk ) ≥ 0] with I[H1 (β, xj , xk ) ≥ 0], I[H0 (β, xj , xk ) ≥ 0] respectively, is
Op (n−1/2 ).
To derive a rate of convergence for the actual estimator, we will derive a rate for:
X
(I[Ĥ1 (β, xj , xk ) ≥ 0] − I[H1 (β, xj , xk ) ≥ 0])Ĥ1 (β, xj , xk )
(B.17)
j6=k
as well as the second ”half” involving H0 (β, xj , xk ). To establish the negligibility of (B.17),
we will first derive a rate of convergence for
X
1
I[Ĥ1 (β, xj , xk ) < 0]I[H1 (β, xj , xk ) ≥ 0]
n(n − 1)
(B.18)
j6=k
uniformly in β in op (1) neighborhoods of β0 . To do so, we decompose
I[Ĥ1 (β, xj , xk ) < 0] = I[Ĥ1 (β, xj , xk ) < −δn ] + I[Ĥ1 (β, xj , xk ) ∈ [−(δn , 0)]
(B.19)
X
1
Ĥ1 (β, xj , xk )I[Ĥ1 (β, xj , xk ) < −δn ]I[H1 (β, xj , xk ) ≥ 0]
n(n − 1) j6=k
(B.20)
√
where δn is a sequence of positive numbers converging to 0 at the rate log n/ n. We aim to
show each of the above indicator functions multiplied by Ĥ1 (β, xj , xk ) and I[H1 (β, xj , xk ) ≥
0] corresponds to a negligible term, uniformly in β in op (1) neighborhoods of β0 . First,
dealing with
Noting that since Ĥ1 (β, xj , xk ) is bounded uniformly in β, xj , xk it will suffice to show the
negligibility of
X
1
I[Ĥ1 (β, xj , xk ) < −δn ]I[H1 (β, xj , xk ) ≥ 0]
n(n − 1) j6=k
(B.21)
As we did before, we will add and subtract the conditional expectation of the term inside the
double summation. First dealing with the subtraction, we now have a centered U-process,
Khan and Tamer 34
where the variance of the term in the double summation vanishes at the rate of o(1). Thus,
by Theorem 3 in Sherman(1994b) the centered process is uniformly op (n−1 ) Now, dealing
with the addition of the conditional expectation:
X
1
E[I[Ĥ1 (β, xj , xk ) < −δn ]I[H1 (β, xj , xk ) ≥ 0]|xj , xk ]
(B.22)
n(n − 1) i6=j
we can follow the same arguments as in our consistency proof to conclude that it is uniformly
op (n−1 ).
It thus remains to show the negligibility of
X
1
I[Ĥ1 (β, xj , xk ) ∈ [−δn , 0)]Ĥ1 (β, xj , xk )I[H1 (β, xj , xk ) ≥ 0]
n(n − 1) i6=j
(B.23)
Note the absolute value of the above summation is bounded above by
δn
C]
X
1
I[Ĥ1 (β, xj , xk ) ∈ [−δn , 0)]I[H1(β, xj , xk ) ≥ 0]
n(n − 1) i6=j
(B.24)
In the above double summation, we add the sum of indicators I[[xj , xk ] ⊂ C]+I[[xj , xk ] 6⊂
First we evaluate the rate for
X
1
I[Ĥ1 (β, xj , xk ) ∈ [−δn , 0)]I[H1(β, xj , xk ) ≥ 0]I[[xj , xk ] ⊂ C]
δn
n(n − 1) i6=j
(B.25)
(Similar arguments can be used for the subset [xj , xk ] 6⊂ C, but we omit the details.) We
again add and subtract the conditional (on xj , xk ) expectation of the term inside the above
double summation; the centered process (i.e. after subtracting the conditional expectation)
is op (n−1 ) uniformly in op (1) neighborhoods of β0 after we multiply by δn . Now, regarding
the expectation
E[I[Ĥ1 (β, xj , xk ) ∈ [−δn , 0)]I[H1(β, xj , xk ) ≥ 0]I[[xj , xk ] ⊂ C]]
we can decompose I[H1 (β, xj , xk ) ≥ 0] into I[H1 (β, xj , xk ) = 0] + I[H1 (β, xj , xk ) > 0]. We
first deal with the term:
E[I[Ĥ1 (β, xj , xk ) ∈ [−δn , 0)]I[H1(β, xj , xk ) = 0]I[[xj , xk ] ⊂ C]]
Note that the indicator I[H1 (β, xj , xk ) = 0] is 0 unless β = β0 as [xj , xk ] ⊂ C, and note we
have
E[I[Ĥ1 (β0 , xj , xk ) ∈ [−δn , 0)] = op (n−1 )
Khan and Tamer 35
Therefore it only remains to establish a rate for
E[I[Ĥ1 (β, xj , xk ) ∈ [−δn , 0)]I[H1(β, xj , xk ) > 0]I[[xj , xk ] ⊂ C]]
We will simply expand the term E[I[H1 (β, xj , xk ) > 0]I[[xj , xk ] ⊂ C]] as a function of
β around β0 - the lead term is 0, and the remainder term is O(kβ − β0 k) which is op (1).
However this will not suffice to conclude that the estimator is root-n consistent, only that is
√
o( δn ). But by a second application of Theorem 1 in Sherman(1994b), this time looking in
√
√
o( δn ) neighborhoods of β0 we can conclude the estimator is indeed n-consistent.
Now that root-n consistency has been established we can apply Theorem 2 in Sherman(1994b) to attain asymptotic normality. A sufficient condition is that uniformly over
√
Op (1/ n) neighborhoods of β0 ,
1
1
1
Gn (β) − Gn (β0 ) = (β − β0 )′ V (β − β0 ) + √ (β − β0 )′ Wn + op ( )
2
n
n
(B.26)
where Wn converges in distribution to a N(0, Ω) random vector, and V is positive definite.
In this case the asymptotic variance of β̂ − β0 is V −1 ΩV −1 .
We will turn to (B.26). Here, we will again work with the U-statistic decomposition in,
for example, Serfling(1980) as our objective function is a third order U-process. We will first
derive an expansion for G(β) around G(β0 ), since G(β) is related to the limiting objective
function. We note that even though Gn (β) is not differentiable in β, G(β) is sufficiently
smooth for Taylor expansions to apply by Assumptions A4, D2. Taking a second order
expansion of G(β) around G(β0 ), we obtain
1
G(β) = G(β0 ) + ∇β G(β0 )′ (β − β0 ) + (β − β0 )′ ∇ββ G(β ∗ )(β − β0 )
2
(B.27)
where ∇β and ∇ββ denote first and second derivative operators and, and β ∗ denotes an
intermediate value. We note that the first two terms of the right hand side of the above
equation are 0, the first by how we defined the objective function, and the second by our
identification result in Theorem 1. We will thus show the following result:
∇ββ G(β ∗ ) = V + op (1)
(B.28)
The form of the matrix V is as the second derivative with respect to β of the following
function evaluated at β = β0 .
E[H1 (β, xj , xk )I[H1 (β, xj , xk ) ≥ 0]] − E[H0 (β, xj , xk )I[H0 (β, xj , xk ) ≤ 0]]
(B.29)
Khan and Tamer 36
Note that by definition:
H1 (β, xj , xk ) =
H0 (β, xj , xk ) =
Z
Z
xk
xj
1
P c ≥ u β; ǫ ≥ u (β − β0 )|u) −
2
xk
xj
′
1
−
2
Z
′
u′ (β−β0 )
Z
fX (u)du
!
f(c,ǫ)|u (c, e)dcde fX (u)du
e+xβ0
So, the second derivative is:
V = E [∇ββ H1 I1 + 2∇β H1 ∇β I1 + H1 ∇ββ I1 − ∇ββ H0 I0 − 2∇β H0 ∇β I0 − H0 ∇ββ I0 ] (B.30)
where I1 = I[H1 (xj , xk ; β0 ) ≥ 0] and similarly for I0 . The above expression for V can be
simplified. For example, we have ∇ββ H1 (xj , xk ; β) − ∇ββ H0 (xj , xk ; β) = 0 on the set C.
Next, notice that by a simple integration by parts argument,
E[∇β H1 ∇β I1 + H1 ∇ββ I1 ] = 0
(B.31)
and similarly for its H0 part. Hence, what remains is
V
= E [∇β H1 ∇β I1 − ∇β H0 ∇β I0 ]
"
Z
xk
= 2E 1[[xj , xk ] ⊂ C]
xfǫ (0|x)dFx
xj
Z
= 2E[I[[xj , xk ] ⊆ C]G(xj , xk )G(xj , xk )′ ]
xk
xj
x′ fǫ (0|x)dFx
#
(B.32)
(B.33)
(B.34)
We next turn attention to the deriving the form of the outer product of the score term
in Theorem 2 in Sherman(1994b). Note this was basically done in our arguments showing
root-n consistency. This involves the conditional expectation, conditioning on each of the
three arguments in the third order process, subtracting the unconditional expectation. We
first condition on the first argument, denoted by the subscript j. Note here we are taking the expectation of the term I[vl ≥ x′l β] − 21 as well as 21 − dl I[vl ≤ x′l β], so using the
same arguments as we did for the unconditional expectation, the average of this conditional
√
expectation is Op (kβ−β0 k2 )/ n, and thus asymptotically negligible for β in Op (n−1/2 ) neighborhoods of β0 . The same applies to the expectation conditional on the second argument of
the third-order U-process, denoted by the subscript k.
We therefore turn attention to expectation conditional on the third argument, denoted
by the subscript l. Here we proceed as before when showing root-n consistency, expanding
I[H1 (β, xj , xk ) ≥ 0]I[vl ≥ x′l β] −
1
2
(B.35)
Khan and Tamer 37
around β = β0 . Recall this yielded the mean 0 process:
n
1
1X
E[G(xj , xk )Iljk ](I[vl ≥ x′l β0 ] − )
n l=1
2
(B.36)
plus a negligible remainder term. Consequently, using the same arguments of the half of the
objective function involving H0 (·, ·, ·) we can express the linear term in our expansion (used
to derive the form of the outer score term) as:
n
1X
E[G(xj , xk )Iljk ](I[vl ≥ x′l β0 ] − dl I[vi ≤ x′l β0 ])′ (β − β0 ) + op (n−1 )
n l=1
(B.37)
which corresponds to
1
√ (β − β0 )′ Wn
n
(B.38)
′
Wn ⇒ N(0, E[δ0l δ0l
])
(B.39)
where
δ0l = E[G(xj , xk )Iljk ](I[vl ≥ x′l β0 ] − dl I[vi ≤ x′l β0 ])
(B.40)
This completes a representation for the linear term in the U-statistic representation. The
remainder term, involving second and third order U-processes (see, e.g. equation (5) in
Sherman(1994b)), can be shown to be asymptotically negligible (specifically it is op (n−1 )
uniformly in β in an Op (n−1/2 ) neighborhood of β0 using Lemma 2.17 in Pakes and Pollard
(1989) and Sherman(1994b) Theorem 3).
Combining this result with our results for the Hessian term, and applying Theorem 2 in
Sherman(1994b), we can conclude that
√
n(β̂ − β0 ) ⇒ N(0, V −1 ΩV −1 )
(B.41)
where
′
Ω = E[δ0l δ0l
]
which establishes the proof of the theorem.
Khan and Tamer 38
TABLE I
Simulation Results for Censored Regression Estimators
CI Censoring, Homosked. Errors
Mean Bias
α
Med. Bias
50 obs.
MD
HKP
Buckley James
YJW
0.2411
-0.0322
-0.0296
-0.2163
0.1159
-0.0356
-0.0437
-0.2288
0.6098
0.2746
0.2124
0.3106
0.1963
0.2182
0.1461
0.2534
-0.1617
0.0009
0.0025
0.0789
-0.0695
0.0051
-0.0049
0.0655
0.7169
0.1600
0.1245
0.1555
0.1948
0.1249
0.0693
0.1158
100 obs.
MD
HKP
Buckley James
YJW
0.1059
-0.0111
-0.0235
-0.1524
0.0594
-0.0112
-0.0273
-0.1441
0.2979
0.1744
0.1299
0.2283
0.1232
0.1375
0.0918
0.1839
-0.0824
0.0035
0.0061
0.0527
-0.0408
0.0017
0.0052
0.0370
0.2781
0.0948
0.0794
0.1058
0.1106
0.0754
0.0535
0.0783
200 obs.
MD
HKP
Buckley James
YJW
0.0438
0.0015
-0.0210
-0.0934
0.0368
-0.0040
-0.0240
-0.0776
0.1711
0.1332
0.1048
0.1655
0.0935
0.1051
0.0738
0.1293
-0.0235
-0.0011
0.0083
0.0300
-0.0126
-0.0023
0.0091
0.0276
0.1328
0.0686
0.0525
0.0696
0.0735
0.0545
0.0347
0.0532
400 obs.
MD
HKP
Buckley James
YJW
0.0095
-0.0048
-0.0047
-0.0391
0.0003
-0.0142
-0.0048
-0.0358
0.1056
0.0945
0.0725
0.1077
0.0626
0.0736
0.0492
0.0835
-0.0097
0.0042
0.0002
0.0129
-0.0088
0.0032
-0.0009
0.0116
0.0808
0.0492
0.0369
0.0467
0.0492
0.0383
0.0253
0.0358
RMSE
MAD
Mean Bias
β
Med. Bias
RMSE
MAD
Khan and Tamer 39
TABLE II
Simulation Results for Censored Regression Estimators
CI Censoring, Heterosked. Errors
Mean Bias
α
Med. Bias
50 obs.
MD
HKP
Buckley James
YJW
0.3083
2.2080
1.9799
-0.1381
0.1576
0.0252
0.6611
-0.1663
0.9230
31.5952
5.5605
0.4222
0.2911
2.5406
0.6679
0.3352
0.0156
-0.8822
-1.9703
-0.0257
-0.0072
-0.0768
-0.9228
0.0435
0.9642
10.4137
4.3843
0.6394
0.3599
1.3964
0.9757
0.4794
100 obs.
MD
HKP
Buckley James
YJW
0.1642
4.8267
2.5409
-0.1266
0.0751
-0.0131
0.8277
-0.1028
0.5633
86.1774
8.3142
0.3794
0.1605
5.0571
0.8277
0.2621
-0.0231
-0.8983
-2.4717
0.0354
-0.0055
-0.0609
-1.0941
0.0556
0.6197
14.1936
7.1011
0.5035
0.2626
1.2946
1.0941
0.3685
200 obs.
MD
HKP
Buckley James
YJW
0.0619
0.0665
4.6481
-0.0788
0.0456
0.0133
1.2122
-0.0676
0.2194
0.4674
34.5088
0.2301
0.1245
0.2301
1.2122
0.1713
-0.0025
-0.0520
-3.8331
0.0352
0.0088
0.0012
-1.4037
0.0285
0.2847
0.4418
22.0628
0.3415
0.1673
0.3236
1.4037
0.2543
400 obs.
MD
HKP
Buckley James
YJW
0.0201
-0.0038
4.9056
-0.0454
0.0072
-0.0206
1.5762
-0.0493
0.1340
0.1888
21.8564
0.1714
0.0808
0.1431
1.5762
0.1323
-0.0134
0.0180
-4.2223
0.0282
-0.0165
0.0158
-1.7588
0.0294
0.2055
0.2979
16.1604
0.2697
0.1378
0.2347
1.7588
0.2017
RMSE
MAD
Mean Bias
β
Med. Bias
RMSE
MAD
Khan and Tamer 40
TABLE III
Simulation Results for Censored Regression Estimators
CD Censoring, Homosked. Errors
Mean Bias
α
Med. Bias
50 obs.
MD
HKP
Buckley James
YJW
0.2318
0.1195
-0.0390
0.8236
0.1273
0.0230
-0.0477
0.6180
0.6043
0.6267
0.2693
1.3345
0.2937
0.3883
0.1737
0.8776
0.2704
-0.2123
-0.0021
-2.2428
0.0679
-0.1960
0.0007
-1.9628
1.0035
0.6009
0.2846
2.7286
0.3673
0.4385
0.1394
2.2450
100 obs.
MD
HKP
Buckley James
YJW
0.1151
0.1042
-0.0355
0.7568
0.0609
0.0457
-0.0459
0.7295
0.3790
0.3656
0.1779
2.5556
0.1681
0.2526
0.1312
0.9934
0.1739
-0.1984
0.0050
-2.0231
0.0960
-0.1837
-0.0004
-1.9468
0.5459
0.3914
0.1779
3.1495
0.2419
0.3039
0.0787
2.2189
200 obs.
MD
HKP
Buckley James
YJW
0.0561
0.0799
-0.0187
0.8143
0.0342
0.0602
-0.0265
0.7990
0.2189
0.2199
0.1216
0.9926
0.1207
0.1700
0.0788
0.8206
0.1246
-0.1817
0.0057
-1.9930
0.0740
-0.1778
0.0032
-1.9612
0.3580
0.2753
0.1071
2.0831
0.1947
0.2247
0.0420
1.9930
400 obs.
MD
HKP
Buckley James
YJW
0.0314
0.0625
-0.0061
0.9022
0.0215
0.0590
-0.0107
0.8839
0.1579
0.1525
0.0838
0.9728
0.0998
0.1202
0.0581
0.9029
0.0642
-0.1811
0.0016
-2.0317
0.0391
-0.1859
0.0013
-1.9860
0.2268
0.2368
0.0620
2.0777
0.1114
0.2013
0.0234
2.0317
RMSE
MAD
Mean Bias
β
Med. Bias
RMSE
MAD
Khan and Tamer 41
TABLE IV
Simulation Results for Censored Regression Estimators
CD Censoring, Heterosked. Errors
Mean Bias
α
Med. Bias
50 obs.
MD
HKP
Buckley James
YJW
0.3221
17.3471
1.7253
0.9745
0.1483
0.4300
0.3113
0.5735
0.8822
153.2266
5.6605
2.5513
0.3214
17.4901
0.3716
1.0331
0.4476
-4.7159
-2.3871
-2.7026
0.1939
-1.0175
-0.9728
-2.1897
1.6561
27.6868
5.5402
4.2208
0.4751
4.8081
0.9790
2.7121
100 obs.
MD
HKP
Buckley James
YJW
0.1452
20.1083
2.3266
0.7854
0.0492
0.3676
0.4856
0.6658
0.4698
163.0567
7.4144
3.6963
0.2012
20.1872
0.4856
1.0380
0.2691
-4.4281
-3.3150
-2.3131
0.1049
-0.8931
-1.1578
-2.1072
0.7333
24.1675
9.8721
4.7598
0.2933
4.4517
1.1578
2.5239
200 obs.
MD
HKP
Buckley James
YJW
0.0674
13.3405
4.6914
0.9670
0.0442
0.3622
0.6757
0.7625
0.2425
101.3273
28.3437
3.5172
0.1382
13.3764
0.6757
0.9707
0.1570
-3.2912
-5.4411
-2.4190
0.0872
-0.9041
-1.3982
-2.1862
0.4477
16.3618
32.3954
4.7660
0.2430
3.2944
1.3982
2.4190
400 obs.
MD
HKP
Buckley James
YJW
0.0294
83.9411
6.4436
0.8527
0.0142
0.2946
1.3839
0.8572
0.1827
663.8925
30.7388
0.9087
0.1045
83.9501
1.3839
0.8534
0.1253
-11.1658
-5.8856
-2.2082
0.0862
-0.8572
-2.1991
-2.1888
0.3388
76.6133
19.2469
2.2396
0.1779
11.1658
2.1991
2.2082
RMSE
MAD
Mean Bias
β
Med. Bias
RMSE
MAD
Khan and Tamer 42
TABLE V
Simulation Results for Censored Regression Estimators
End. Censoring, Homosked. Errors
Mean Bias
α
Med. Bias
50 obs.
MD
HKP
Buckley James
YJW
0.3221
-0.2837
-0.6154
1.3391
0.1483
-0.3124
-0.6246
1.1141
0.8822
1.0604
0.9891
2.2510
0.3214
0.6562
0.7096
1.1936
0.4476
-0.1277
-0.2267
1.6087
0.1939
-0.1108
-0.2375
1.6710
1.6561
0.3055
0.3080
1.9920
0.4751
0.1899
0.2419
1.6710
100 obs.
MD
HKP
Buckley James
YJW
0.1452
-0.4087
-0.6412
1.3479
0.0492
-0.3632
-0.6571
1.0610
0.4698
0.8266
0.8280
2.2394
0.2012
0.5193
0.6647
1.1183
0.2691
-0.1684
-0.2331
1.5984
0.1049
-0.1586
-0.2285
1.6145
0.7333
0.2587
0.2723
1.9851
0.2933
0.1689
0.2316
1.6145
200 obs.
MD
HKP
Buckley James
YJW
0.0674
-0.4616
-0.6280
10.5304
0.0442
-0.4471
-0.6255
1.3098
0.2425
0.6359
0.7268
175.7172
0.1382
0.4801
0.6255
1.3183
0.1570
-0.1907
-0.2310
3.4390
0.0872
-0.1877
-0.2346
1.1062
0.4477
0.2241
0.2505
39.9520
0.2430
0.1877
0.2346
1.1190
400 obs.
MD
HKP
Buckley James
YJW
0.0294
-0.4656
-0.6304
4.2454
0.0142
-0.4236
-0.6229
1.4062
0.1827
0.5817
0.6765
26.2483
0.1045
0.4236
0.6229
1.4534
0.1253
-0.1906
-0.2310
1.8082
0.0862
-0.1809
-0.2321
-0.7231
0.3388
0.2125
0.2405
6.7396
0.1779
0.1809
0.2321
0.7727
RMSE
MAD
Mean Bias
β
Med. Bias
RMSE
MAD
Khan and Tamer 43
TABLE VI
Empirical Study of Drug Relapse Data
Weibull
Log Log.
LAD
YJW
HKP
Buck. James
Min. Dis.
INT
4.8350
(0.1187)
3.8752
(0.1110)
3.8368
(0.1070)
3.8620
(0.1457)
3.4108
(0.3533)
3.7295
(0.1310)
3.6130
(0.1411)
SITE
-0.4866
(0.1040)
-0.5254
(0.0938)
-0.4926
(0.0959)
-0.4952
(0.0944)
-0.3547
(0.1352)
-0.5559
(0.1016)
-0.2578
(0.1176)
IV
-0.3673
(0.0985)
-0.1835
(0.0862)
-0.1769
(0.0876)
-0.1683
(0.0896)
-0.1192
(0.0943)
-0.1277
(0.0908)
-0.0919
(0.1007)
NDT
-0.0243
(0.0078)
-0.0209
(0.007)
-0.0119
(0.0075)
-0.0122
(0.0062)
-0.0164
(0.0065)
-0.0186
(0.0071)
-0.0393
(0.0065)
RACE
0.2964
(0.1073)
0.3288
(0.0952)
0.3393
(0.0948)
0.3292
(0.1131)
0.4411
(0.1147)
0.3413
(0.0990)
0.4050
(0.0987)
TREAT
0.4215
(0.0905)
0.6114
(0.0839)
0.6243
(0.0820)
0.6075
(0.0919)
0.7605
(0.1451)
0.6120
(0.0847)
0.7952
(0.1221)
FRAC
1.1543
(0.0990)
1.468
(0.0839)
1.2488
(0.0798)
1.2357
(0.0938)
1.6790
(0.3038)
1.5732
(0.0869)
1.5412
(0.1123)
Khan and Tamer 44
TABLE VII
Empirical Study of Selective Compliance using Drug Relapse Data
OLS
2SLS
MD
MDIV
INT
4.3726
(0.0807)
4.3090
(0.2141)
4.0869
(0.1827)
4.5012
(0.1047)
IV
-0.1783
(0.0777)
-0.1863
(0.0812)
-0.1219
(0.0668)
-0.1457
(0.0551)
RACE
0.2840
(0.0836)
0.2651
(0.0879)
0.3087
(0.0461)
0.3391
(0.0645)
NDT
-0.0171
(0.0067)
-0.0177
(0.0071)
-0.0313
(0.0171)
-0.0285
(0.0082)
SITE
-0.4187
(0.0833)
-0.2354
(0.1230)
-0.3123
(0.0932)
-0.2382
(0.1183)
LOS
0.0086
(0.0005)
0.0050
(0.0018)
0.0114
(0.0013)
0.0067
(0.0009)